Parallel Axis Theorem

Imagine two axes that are parallel. One passes through the center of mass (CM) of an object of mass M whose moment of inertia about the CM is I_CM. The other axis passes through some point P about which the object has moment of inertia I_P. The distance between the axes is h. The theorem states that"

For example - the moment of inertia of a rod about an axis through the center is ML²/12 and about an axis through one end is ML²/3 and the separation between the two axis is L/2

The origin of coordinates is at the CM. Now we want to compute the moment of inertia about point P so we sweep around the entire mass and each "chunk" of mass dm is located by vector r measured from point P. Point P is given by coordinates a and b and the distance between the CM and point P is h where:

1. The first term is just the moment of inertia about the center of mass
2. In the second term the h2 is constant - pull out of the integral and the integral integrates to total mass M
3. The third and fourth terms separately integrate to zero because x and y are measured from the center of mass so by definition these integrals vanish.

Theorem proved.